As a next step, we give embedded pairs for SSPERK\((s,3)\)  methods. Kraaijevanger made the first step to determine the SSPERK\((s,3)\)  methods. Namely, he treated the SSPERK\((4,3)\)  case [1]}. Later the SSPERK\((s,3)\)  methods were characterized, where \(s=n^2\)  and \(n\ge 2\)  is an integer [2]}. The corresponding SSP coefficients are \(n^2-n=s-\sqrt{s}\) . The Butcher form of SSPERK\((s,3)\)  methods is given (REF )-(REF ).
\(A = \begin{tikzpicture}[baseline={(m.center)}][matrix of math nodes,left delimiter=(,right delimiter=)] (m){0 & & & & & & & \\\frac{1}{n(n-1)} & & & & & & & \\\frac{1}{n(n-1)} & \ddots & & & & & & \\\frac{1}{n(n-1)} & & \ddots & & & & & \\\vdots & & & \ddots & & & & & & \\\vdots & & & & \ddots & & & \\\frac{1}{n(n-1)} & \ldots & \frac{1}{n(n-1)} & \ldots & \ldots & \frac{1}{n(n-1)} & & & & \\\frac{1}{n(n-1)} & & \frac{1}{n(n-1)} & \frac{1}{n(2n-1)} & \ldots & \frac{1}{n(2n-1)} & \frac{1}{n(n-1)} & & & \\\vdots & & \vdots & \vdots & & \vdots & \vdots & \ddots \\\makebox{[}0pt][l]{\smash{\underbrace{\phantom{\begin{matrix}\frac{1}{n(n-1)} & \ldots &\end{matrix}}}_{\text{$\frac{(n-2)(n-1)}{2}$}}}}\frac{1}{n(n-1)} & \ldots & \frac{1}{n(n-1)} &\frac{1}{n(2n-1)} & \ldots & \frac{1}{n(2n-1)} & \makebox{[}0pt][l]{\smash{\underbrace{\phantom{\begin{matrix}\frac{1}{n(n-1)} & \ldots &\end{matrix}}}_{\text{$\frac{n(n-1)}{2}-1$}}}}\frac{1}{n(n-1)} & \ldots & \frac{1}{n(n-1)} & 0 \\};[loosely dotted] (m-1-1)-- (m-10-10);(m-8-4.north west) rectangle (m-10-6.south east);\end{tikzpicture}\in \mathbb {R}^{n^2\times n^2},\) 
